System and method for halftoning using a parametrically controlled hexagonal halftone dot shape threshold function

ABSTRACT

Methods and apparatuses for halftoning an image are provided using a parametrically controlled hexagonal halftone dot shape threshold function that reduces tone reproduction irregularities in the halftoned image which can occur at darker gray levels. The halftoning transforms image data representing contone image pixels into halftoned image data in the form of clustered-dot hexagonal halftone screens for representing halftone dots of a halftoned image. Weight parameters can be used to control the rate at which a respective vertex of a halftone dot approaches a vertex of a neighboring halftone dot in relation to gray level. The hexagonal halftone dot shape threshold function can also control the shape of the sides of the halftone dots.

BACKGROUND

The presently disclosed embodiments are directed toward methods andsystems for printing, reproducing or displaying images. Moreparticularly, the teachings disclosed herein are applicable to methodsand apparatuses wherein clustered-dot halftoning is implemented.

Digital images may be formatted as contone (continuous tone) imageshaving a wide range of tonal values or may be formatted as coarselyquantized images having a limited number of tonal values, such as twolevels for a binary image. Digital halftoning is a process oftransforming a contone image to a coarsely quantized image. Digitalhalftoning is an important step in printing or displaying digital imagespossessing contone color tones because most printing processes areoperating in a binary mode. Examples of such marking processes areoffset printing presses, xerography, and ink-jet printing. In theseprocesses, for each color separation of an image, a correspondingcolorant spot is either printed or not printed at any specified imagelocation, or pixel. Digital halftoning controls the printing of colordots formed by combinations of colorant spots of a colorant set, wherethe spatial averaging of the printed colorant dots, such as by the humanvisual system, provides the illusion of the required continuous tones.

Digital images and the resulting prints are formed of one or morecolorant separations, also referred to as “color separations.” Amonochrome image is formed of one colorant separation, typically black.Process color images are typically constructed of cyan, magenta, yellow,and black separations. Duotone and tritone images, are formed of two andthree separations, respectively. Spot color images have multiplecolorant separations, where at least one colorant is positionedspatially nonoverlapping with other colorants. Extended colorant setimages typically include the process-color colorant separations plus oneor more additional colorant separations such as green, orange, violet,red, blue, white, varnish, light cyan, light magenta, gray, dark yellow,metallics, and so forth. In the present teachings, we will use the terms“color images”, “color dots”, “color spots”, “colorant” and similarlanguage to refer to images and marking systems with any number ofcolorants. The teachings herein apply particularly to any individualcolor separation of a digital image and resulting print, where thatdigital image or print can be composed of one or more separations. Withthe advent of computers, it is desirable for graphic artists and othersto manipulate contone images and print them as halftone images. However,typical computer printers and typesetters are incapable of printingindividual halftone dots in an infinite number of sizes. Instead, eachhalftone dot of a printed picture is in turn comprised of a collectionof discrete, smaller “spots” or “pixels”, which are generally thesmallest marks a printer or typesetter can make.

FIG. 1 illustrates how one such dot may be made up of individual pixels.A grid 100 is comprised of a set of 100 contiguous pixels, andtherefore, is capable of representing 101 shades, or levels, of grayfrom totally light and white (0% gray level and no pixels darkened) tototally dark and black (100% gray level and all pixels darkened). Forevery 1% increase in darkness, one pixel in the set will be darkened, orturned on for printing. For example, at 1% gray, a single pixel 1 isdarkened. At 2% gray, pixel 2 is darkened as well, so that the dot iscomprised of two pixels. At 10% gray, ten pixels 1-10 are darkened.Pixels are either dark and thus printed, or not dark and not printed,and are not individually capable of representing shades of gray.

A common halftone technique is called screening, which compares therequired continuous color tone level of each pixel for each colorseparation with one or more predetermined threshold levels. Thepredetermined threshold levels are typically defined for halftone cellsthat are tiled to fill the plane of an image, thereby forming a halftonescreen of threshold values. At a given pixel, if the required color tonelevel is greater than the halftone threshold level for that pixel, a “1”is generated in the halftone output, so that a colorant spot is printedat that specified pixel in the subsequent printing operation. If therequired color tone at a given pixel is less than the halftone thresholdlevel for that pixel, a “0” is generated in the halftone output, so thata colorant spot is not printed at that specified pixel in the subsequentprinting operation. The output of the screening process is a binarypattern that controls the printing of multiple small spots or pixelsthat are printed. The printed spots can be grouped or “clustered” toform print structures that are relatively stable for a given printingprocess. We refer to these clusters as “clustered-dots” or “dots”, andthey are regularly spaced as determined by the size, shape, and tilingof the halftone cell. Conventional periodic halftone screens andhalftone screen outputs can be considered as two-dimensional repeatedpatterns, possessing two fundamental spatial frequencies, which arecompletely defined by the geometry of the halftone screens.

When halftoning using screening, rather than darkening a random pixel ina cell for every increase in gray level, it is preferable for the pixelsto be darkened in a specific order pursuant to a dot shape function,also known as a dot function or spot function. The order of darkeningpixels in FIG. 1 reflects a dot shape function which attempts tomaintain a generally compact shape for the dot as it increases in sizeto represent increasing gray levels. Topological curves (labeled from10% to 100% in increments of 10%) generally define the outline of thedot as it increases in gray level. At 10% gray, the darkened pixelssubstantially conform with the 10% curve, and form a nearly circular dotaround the center 110 of the dot shape function. At 20% gray, thedarkened pixels form a slightly larger dot which substantially conformswith the 20% curve. At 50% gray, the pixels form a dot which is insubstantially the shape of a diamond. The diamond shape in this exampleis an attempt to control the way in which neighboring dots touch. Thegray level of printed dots tends to be printed with a more consistentdarkness if the dot touch points are optimized for the given markingprocess. It is common to optimize the touch points by shaping the dotsto touch at corners rather than boundaries of circles or other shapes,but certain marking processes may require other shape optimization, suchas straight side touching or curved boundary touching. At 90% gray,nearly all of the pixels are darkened except for 10 pixels 91-100, thesepixels being evenly dispersed at the four corners of the dot.

It is often desirable to store the pixels' representation of the dotshape function in memory for later use. To do so, the dot shape functionis evaluated at the location of each pixel in the cell, the pixels arerank ordered according to their respective dot shape function values,and a threshold value from 0% to 100% is assigned to each pixelaccording to its rank. The values are often stored in bit form, such as0 to 255 for an 8 bit system. Where dot shape function values areidentical or nearly identical (within roughly 10%) for multiple pixelsin a dot, their order can be determined by any of a number of secondaryconsiderations. For, instance a marking process or imager may markpixels in a more consistent manner if pixels are preferentially added toa side, such as the lead edge, or trail edge of the dot as it movesthrough the process or start-of-scan or end-of-scan aide of a dotrelative to a laser imager scanning direction. Angular considerationsare sometimes used to rank pixels. For instance, to have minimaldisplacement of the centroid of the dot from gray level to gray level,pixels with nearly identical dot shape function values are sometimesselected by spiraling around the dot in quadrant steps. As anotherexample, printed dot consistency is sometimes achieved by preferentiallygrowing a dot in a vertical or horizontal direction where pixels havingnearly identical dot shape function values are ranked to provide moregrowth in the preferred direction. In some cases, the fill order forpixels of nearly identical dot shape function values could be random, orselected by any of a number of other criteria. In this way, each pixelhas an associated “threshold value” in the halftone screen which isequal to the gray level at which that pixel is darkened in the printedimage.

Referring to FIG. 2, if a dot which represents a gray level of 75% isdesired, the dot is created by darkening every pixel with a thresholdvalue of 75% or less. A 75% gray-level dot is indicated by outline 200.A single collection of pixels with threshold values representing a dotshape function can be referred to as a “halftone cell” or “cell.” Asused herein, a cell is a quantized representation of a dot shapefunction, and for a given cell, the threshold values of the pixelswithin the cell map the dot shape function.

In this manner, the “digital screen” is created, as an array of cellswith pixels having threshold values. Each pixel has a set position and aset threshold value within the cell. Likewise, each cell has a setposition within the digital screen. To create a halftone image, acontone image is broken down into an array of pixel-sized samples, andthe gray level of each contone sample is stored. Next, each contonesample is compared with the halftone threshold value of thecorresponding pixel in the halftone screen, and the pixel is darkened inthe subsequent print image if the gray level of the contone sample isgreater than the threshold value for that pixel. All the pixels of thedigital screen are at set positions with respect to one another, suchthat a contone sample from the “top-left” of the picture would becompared with a pixel at the “top-left” of the digital screen. In otherwords, each digital screen pixel has a position which corresponds withand is associated with a position on the original contone picture.

FIG. 3 illustrates a portion of a halftone image 300 represented by dots310. FIG. 3 was created by comparing an input image having a spatiallyconsistent 5% gray value to a digital screen containing a 3-by-3 arrayof cells. Each cell contained 100 pixels, and only the pixels withthreshold values of 5% or less were darkened and printed. Accordingly, a3-by-3 array of dots was created, each dot having five pixels. FIG. 4was created by comparing an input image having a spatially consistent95% gray value to the same digital screen. All the pixels with grayvalues of 95% or less were darkened and printed to form a halftone image400. Although the resulting halftone image 400 is really comprised ofnine large dots 410, the naked eye perceives the halftone image as beingsmaller white dots 420 on a field of black. The white dots are groups ofthe pixels that are formed from halftone screen values having highthresholds. The white dots can be formed from one or more cells.

Halftoning attempts to render images to printable form while avoidingunwanted visual texture, known as moiré, and tone reproductionirregularities. The two key aspects of halftone screen design are thegeometry of periodic dot placement and the shape of the halftone dots.Controlling halftone dot shape has been a lower priority in laserprinters because printer pixel resolution, typically measured in rastersper inch referring to the number of smallest printable spots per unitlength, has been too low. Consider, for example, the task of controllingdot shape of a 212 cell per inch (cpi) halftone screen used with aprinter having a resolution of 600 rasters/inch, where the halftone cellis only two rasters in height. As laser printing resolutions reach 2400rasters/inch, and greater, controlling halftone dot shape provides agreater impact in improving a printed image.

Hexagonal halftones have been used for process-color printing to avoidmoiré that can occur with conventional halftone geometries. Inparticular, hexagonal dot geometries have been used to reduce moirébetween yellow screens and cyan or magenta screens at conventionalangles, such as taught by U.S. Pat. No. 5,381,247 for “Method forReducing 2-Color Moiré in 4-Color Printing” to C. Hains, which is herebyincorporated by reference herein in its entirety. However, this methodhas not been widely adopted since it can create a tone reproductionirregularity, or “bump”, that occurs as the sides of the hexagonal dotsgrow toward each other when producing darker gray levels in thehalftoned image. It is desirable solve this problem with tonereproduction irregularities when using hexagonal halftones to makebetter use of their advantages as taught by US Publication No.2008/0130055 for “Moiré-Free Color Halftone Configuration EmployingCommon Frequency Vectors” to Wang, et al., US Publication No.2008/0130054 for “N-Color Printing with Hexagonal Rosettes” to Wang, etal., and U.S. Pat. No. 6,798,539 for “Method for Moiré-Free ColorHalftoning Using Non-Orthogonal Cluster Screens” to Wang, et al. whichare hereby incorporated by reference herein in their entirety.

BRIEF DESCRIPTION

Methods and apparatuses for halftoning an image are provided using aparametrically controlled hexagonal halftone dot shape thresholdfunction data in the form of clustered-dot hexagonal halftone screens.

The method includes receiving contone image data including pixel valuesrepresenting gray-scale color densities and pixel location information;and halftoning the image using an image processor generatingclustered-dot halftone screen output representing halftone dots bycomparing the image pixel values to a threshold function in the form ofa parametrically controlled hexagonal dot shape function.

In examples of the method of halftoning disclosed herein, theparametrically controlled hexagonal dot shape function isQ ₁ =a ₁ cos(π(2h ₁ /H ₁))+a ₂ cos(π(2h ₂ /H ₂))+a ₃ cos(π(2h ₃ /H ₃))where H₁, H₂ and H₃ are the periods of respective sets of parallel linesintersecting to define a hexagonal halftone screen lattice, h₁, h₂ andh₃ are perpendicular distances from a point of interest in the image toa respective one of the parallel lines from each set, and a₁ a₂ and a₃are weight parameters controlling the rate at which a respective vertexof a halftone dot approaches a vertex of a neighboring halftone dot inrelation to gray level, the halftone dots being substantially circularin halftoned image highlights having lower gray levels, substantiallyhexagonal in halftoned image midtones, and having triangular holesbetween adjacent dots in halftoned image shadows having higher graylevels.

In other examples of the method of halftoning disclosed herein, theparametrically controlled hexagonal dot shape function isQ ₂ =a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H ₂)^(γ) ² )+a ₃cos(π(2h ₃ /H ₃)^(γ) ³ )where γ_(i) controls the shape of the sides of the halftone dots, H₁, H₂and H₃ are the periods of respective sets of parallel lines intersectingto define a hexagonal halftone screen lattice, h₁, h₂ and h₃ areperpendicular distances from a point of interest in the image to aclosest respective one of the parallel lines from each set, and a₁ a₂and a₃ are weight parameters controlling the rate at which a respectivevertex of a halftone dot approaches a vertex of a neighboring halftonedot in relation to gray level, the halftone dots being substantiallycircular in halftoned image highlights having lower gray levels,substantially hexagonal in halftoned image midtones, and havingtriangular holes between adjacent dots in halftoned image shadows havinghigher gray levels.

In other examples of the method of halftoning disclosed herein, theparametrically controlled hexagonal dot shape function isQ ₃=(a ₁ +a ₂ +a ₃)−(a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H₂)^(γ) ² )+a ₃ cos(π(2h ₃ /H ₃)^(γ) ³ ))where γ_(i) controls the shape of the sides of the halftone dots, H₁, H₂and H₃ are the periods of respective sets of parallel lines intersectingto define a hexagonal halftone screen lattice, h₁, h₂ and h₃ areperpendicular distances from a point of interest in the image to aclosest respective one of the parallel lines from each set, and a₁ a₂and a₃ are weight parameters controlling the rate at which a respectivevertex of a halftone dot approaches a vertex of a neighboring halftonedot in relation to gray level, providing a compact growth sequence fromdark triangles at lower gray levels, to dark hexagons as midtones, tolight round holes between adjacent dots in the darker or higher graylevels.

An image halftoner is provided which includes an image processorreceiving contone image data including pixel values representinggray-scale color densities and pixel location information, the imageprocessor generating clustered-dot halftone screen output of halftonedots by comparing the image pixel values to a threshold function in theform of a parametrically controlled hexagonal dot shape function.

In examples of the image halftoner disclosed herein, the parametricallycontrolled hexagonal dot shape function isQ ₁ =a ₁ cos(π(2h ₁ /H ₁))+a ₂ cos(π(2h ₂ /H ₂))+a ₃ cos(π(2h ₃ /H ₃))where H₁, H₂ and H₃ are the periods of respective sets of parallel linesintersecting to define a hexagonal halftone screen lattice, h₁, h₂ andh₃ are perpendicular distances from a point of interest in the image toa respective one of the parallel lines from each set, and a₁ a₂ and a₃are weight parameters controlling the rate at which a respective vertexof a halftone dot approaches a vertex of a neighboring halftone dot inrelation to gray level, the halftone dots being substantially circularin halftoned image highlights having lower gray levels, substantiallyhexagonal in halftoned image midtones, and having triangular holesbetween adjacent dots in halftoned image shadows having higher graylevels.

In other examples of the image halftoner disclosed herein, theparametrically controlled hexagonal dot shape function isQ ₂ =a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H ₂)^(γ) ² )+a ₃cos(π(2h ₃ /H ₃)^(γ) ³ )where γ_(i) controls the shape of the sides of the halftone dots, H₁, H₂and H₃ are the periods of respective sets of parallel lines intersectingto define a hexagonal halftone screen lattice, h₁, h₂ and h₃ areperpendicular distances from a point of interest in the image to aclosest respective one of the parallel lines from each set, and a₁ a₂and a₃ are weight parameters controlling the rate at which a respectivevertex of a halftone dot approaches a vertex of a neighboring halftonedot in relation to gray level, providing a compact growth sequence fromdark triangles at lower gray levels, to dark hexagons as midtones, tolight round holes between adjacent dots in the darker or higher graylevels.

In other examples of the image halftoner disclosed herein, theparametrically controlled hexagonal dot shape function isQ ₃=(a ₁ +a ₂ +a ₃)−(a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H₂)^(γ) ² )+a ₃ cos(π(2h ₃ /H ₃)^(γ) ³ ))where γ_(i) controls the shape of the sides of the halftone dots, H₁, H₂and H₃ are the periods of respective sets of parallel lines intersectingto define a hexagonal halftone screen lattice, h₁, h₂ and h₃ areperpendicular distances from a point of interest in the image to aclosest respective one of the parallel lines from each set, and a₁ a₂and a₃ are weight parameters controlling the rate at which a respectivevertex of a halftone dot approaches a vertex of a neighboring halftonedot in relation to gray level, providing a compact growth sequence fromdark triangles at lower gray levels, to dark hexagons as midtones, tolight round holes between adjacent dots in the darker or higher graylevels.

A printing device is provided which includes an image processorreceiving contone image data including pixel values representinggray-scale color densities and pixel location information, the imageprocessor generating clustered-dot halftone screen output of halftonedots by comparing the image pixel values to a threshold function in theform of a parametrically controlled hexagonal dot shape function.

In other examples of the printing device disclosed herein, theparametrically controlled hexagonal dot shape function isQ ₁ =a ₁ cos(π(2h ₁ /H ₁))+a ₂ cos(π(2h ₂ /H ₂))+a ₃ cos(π(2h ₃ /H ₃))where H₁, H₂ and H₃ are the periods of respective sets of parallel linesintersecting to define a hexagonal halftone screen lattice, h₁, h₂ andh₃ are perpendicular distances from a point of interest in the image toa respective one of the parallel lines from each set, and a₁ a₂ and a₃are weight parameters controlling the rate at which a respective vertexof a halftone dot approaches a vertex of a neighboring halftone dot inrelation to gray level, the halftone dots being substantially circularin halftoned image highlights having lower gray levels, substantiallyhexagonal in halftoned image midtones, and having triangular holesbetween adjacent dots in halftoned image shadows having higher graylevels; and a print engine rendering a halftoned image by printinghalftone dots having pixels darkened in accordance with their thresholdvalues to represent the appropriate gray scale level of the contoneimage.

In other examples of the printing device disclosed herein, theparametrically controlled hexagonal dot shape function isQ ₂ =a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H ₂)^(γ) ² )+a ₃cos(π(2h ₃ /H ₃)^(γ) ³ )where γ_(i) controls the shape of the sides of the halftone dots, H₁, H₂and H₃ are the periods of respective sets of parallel lines intersectingto define a hexagonal halftone screen lattice, h₁, h₂ and h₃ areperpendicular distances from a point of interest in the image to aclosest respective one of the parallel lines from each set, and a₁ a₂and a₃ are weight parameters controlling the rate at which a respectivevertex of a halftone dot approaches a vertex of a neighboring halftonedot in relation to gray level, the halftone dots being substantiallycircular in halftoned image highlights having lower gray levels,substantially hexagonal in halftoned image midtones, and havingtriangular holes between adjacent dots in halftoned image shadows havinghigher gray levels.

In other examples of the printing device disclosed herein, theparametrically controlled hexagonal dot shape function isQ ₃=(a ₁ +a ₂ +a ₃)−(a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H₂)^(γ) ² )+a ₃ cos(π(2h ₃ /H ₃)^(γ) ³ ))where γ_(i) controls the shape of the sides of the halftone dots, H₁, H₂and H₃ are the periods of respective sets of parallel lines intersectingto define a hexagonal halftone screen lattice, h₁, h₂ and h₃ areperpendicular distances from a point of interest in the image to aclosest respective one of the parallel lines from each set, and a₁ a₂and a₃ are weight parameters controlling the rate at which a respectivevertex of a halftone dot approaches a vertex of a neighboring halftonedot in relation to gray level, providing a compact growth sequence fromdark triangles at lower gray levels, to dark hexagons as midtones, tolight round holes between adjacent dots in the darker or higher graylevels.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a representation of a dot shape function in a halftone cell;

FIG. 2 is a representation of threshold values of pixels of a halftonecell;

FIG. 3 is a portion of a halftone image for an input image having aspatially consistent 5% gray value;

FIG. 4 is a portion of a halftone image for an input image having aspatially consistent 95% gray value;

FIG. 5 is block diagram of an image processing system for generating ahalftoned image using the parametrically controlled hexagonal dot shapefunction described herein;

FIG. 6 is portion of an orthogonal grid illustrating parameters used ingenerating a dot shape threshold function for an orthogonal halftonescreen;

FIG. 7 is a portion of a non-orthogonal grid illustrating parametersused in forming a dot shape threshold function for generating anon-orthogonal halftone screen;

FIG. 8 is a portion of a non-orthogonal grid illustrating parametersused in forming a parametrically controlled hexagonal dot shapethreshold function for generating a hexagonal halftone screen;

FIGS. 9A-9E are gray wedges having varying gray levels printed using theparametrically controlled hexagonal dot shape function described herein,where parameters are varied to control the shape of the sides of thedot;

FIGS. 10A-10I are gray wedges having varying gray levels printed usingthe parametrically controlled hexagonal dot shape function describedherein, where parameters are varied to control the eccentric of the dot;and

FIGS. 11A-11G are gray wedges having varying gray levels printed usingthe parametrically controlled hexagonal dot shape function describedherein, where parameters are varied to control the rotation of the dot.

DETAILED DESCRIPTION

Referring now to FIG. 5, the systems and methods disclosed herein aredirected towards aspects of an image processing system, shown generallyat 500, creating halftone screen output for printing a halftoned imageby print engine 540 of a digital printer 550 a or 550 b. The imageprocessing system 500 includes an image processing halftoner 520transforming image data 518 produced and/or provided by an image source510 into halftoned image data also referred to as halftone screen output524 used to print the image. Examples of the image source 510 caninclude a scanner 511, computer 512, computer network 513, digitalcamera 514, or other image producing machine capable of generating thedigital image data 518. The halftoner 520 can be one or moremicrocontrollers, microprocessors, digital signal processors, centralprocessing units (CPUs), graphical card CPUs, graphical processing units(GPUs), application specific integrated circuits (ASICs), fieldprogrammable gate arrays (FPGAs) and other processor computing devicesimplementing the process of transforming digital image data 518 intohalftone screen output 524 as described herein. In some embodiments, thehalftoner 520 is contemplated as being separate from the printer 550 a,in other embodiments it can be incorporated in the printer 550 b, and inother embodiments the image processing used to generate the halftonescreen output may be distributed among image processors in the printerand image processors separate from the printer.

The image data 518 supplied to the halftoner 520 by the image source510-514, either directly or indirectly, can include image pixels havingpixel values. An “image” as used herein is a pattern of physical lightemitting and/or reflecting and/or absorbing elements that can be printedor displayed for viewing. A digital image is formed of “image data”representing these physical elements, which can be referred to as“pixels.” Image data includes pixel location information, correspondingto the location of the pixels in the image, and pixel valuesrepresenting a grayscale or color density to be produced in the image atthe corresponding location. Pixel values can be represented as a bit ina “binary form” of an image, a gray-scale value in a “grayscale form” ofan image representing the gray level of the image pixel, such as forexample a value falling in the range of 0 to 255 (though others can beused), or a set of color space coordinates in a “color coordinate form”of an image and stored and/or provided in the form of a two-dimensionalarray defining the image.

It is well understood that most digital color printers operate in abinary mode, i.e., for each color separation, a corresponding colorantspot is either printed or not printed at a specified image location orpixel. As described above, digital color halftoning controls theprinting of colorant spots, typically in the a pattern of periodicclustered-dots for each colorant separation, for combinations ofcolorants of a colorant set, where the spatial averaging of the printeddots, such as by the human visual system, provides the illusion of therequired continuous color tones, also referred to as contones. Thepresent systems and methods apply to the processing of color images,wherein each separation is treated, effectively, as a gray-scale orcontinuous tone image for a corresponding colorant in the colorant set.Accordingly, references made herein to the processing of continuous tone(contone), or gray scale, images is intended to refer to the processingof image color separations.

The halftoner 520 evaluates the dot shape function described herein atthe location of each pixel in the contone input image data 518, or aportion thereof, being halftoned to produce the halftone screen output524 for each halftone cell. The pixels of the halftone screen are at setpositions with respect to one another and each halftone screen pixel hasa position which corresponds with and is associated with a position onthe original contone picture. Each halftone cell, represented by itsrespective dot shape function, corresponds to a respective portion ofthe contone image being halftoned and the cells are tiled together torepresent the entire image, or the portions thereof, being halftoned.The input image pixels in each halftone cell are rank ordered accordingto their respective dot shape function values, and a threshold valuefrom 0% to 100% is assigned to each pixel according to its rank in thehalftone screen output 524. The halftone screen output 524 is used torender the halftoned image with print engine 540 by darkening the pixelsin accordance with their threshold values to represent the appropriategray scale level of the contone image.

The halftone screen output 524 can include one or more arrays ofthreshold values, along with 3 parameters—width, height of a givenarray, and an offset of successive rows of the array, as taught by U.S.Pat. No. 4,149,194 for “Variable angle electronic halftone screening” toHolladay, and U.S. Pat. No. 4,185,304 “Electronic Halftone Screening,”also to Holladay, both of which are hereby incorporated herein byreference in their entirety. The halftone screens can be specified inother common formats, such as using angles and frequencies along withthe dot shape function, as taught generally by U.S. Pat. No. 4,196,451for “Electronic Halftone Generator” to Pellar, and U.S. Pat. No.4,149,183 for “Electronic Halftone Generator” also to Pellar, both ofwhich are hereby incorporated herein by reference in their entirety.

The halftone screen output 524 can be used to print the halftoned imageby print engine 540, or it can be stored by a storage device 530 forsubsequent printing. Examples of the storage device 530 can include oneor more of a computer memory, such as random access memory (RAM) orstorage media, such as, magnetic media including, but not limited to,magnetic tapes and discs and optical media such as CD ROM, etc.Furthermore, the storage device 530 may include a computer network fortransmitting output from the image processing halftoner 520 to anotherprocessor, image processing system or rendering device. The printingdevice 540 can include a print engine such as ink-jet print engines,xerographic print engines and electrophotographic print engines.

A known dot shape function, such as the classic Pellar dot shapefunction, also known as the Euclidean profile, is described in U.S. Pat.Nos. 4,196,451 and 4,149,183 to Pellar, incorporated by reference above.It can also be described as a sum of two cosine functions:Q=cos [2π(h ₁ /H ₁)]+cos [2π(h ₂ /H ₂)]  (1)where Q is the threshold function that is compared to image pixel valuesin the halftoning operation. In practice, Q would be scaled to have thesame range as the image data, e.g., [0, 255].

As shown in FIG. 6, in instances where the clustered-dot screen is anorthogonal halftone screen 600, all clusters, or dots, are centered onthe intersection points 602 of a grid, or lattice 604, defined by twosets of parallel lines of which the parallel lines L₁ of the first setare perpendicular to the parallel lines L₂ of the second set. Theperiods of the lattice 604 in two different directions, the 45° and −45°directions in FIG. 6, are defined by the shortest distance between theadjacent parallel lines L₁ of the first group, denoted by H₁ and theshortest distance between the adjacent parallel lines L₂ of the secondgroup, denoted by H₂. The threshold value at an arbitrary point p,defined by the distances, h₁ and h₂, from the point p to the adjacentorthogonal grid lines L₁ and L₂, respectively, is given by the functionQ in Equation (1). It is interesting to note that since the two cosinefunctions in Q of Equation (1) are defined with periods h₁/H₁ and h₂/H₂,respectively, it has been found that it does not matter which particulargrid lines within each of the 2 sets are chosen to define the distanceh₁ or h₂.

Referring now to FIG. 7, an arbitrary non-orthogonal halftone screen 700can be defined by two spatial vectors, V₁ and V₂, as shown. The vectorsV₁ and V₂ define a parallelogram that can completely tile the plane ofthe image being halftoned. The dot shape function can be evaluatedwithin such a parallelogram tile to form a cell. Parallelogram cells canbe converted to other geometric forms, such as Holladay bricks. Ininstances where the clustered-dot screen is a non-orthogonal halftonescreen 700, all clusters are centered on the intersection points 702 ofa grid 704 defined by two sets of parallel lines of which the parallellines L₁ of the first set are not perpendicular to (and not parallel to)the parallel lines L₂ of the second set. The dot shape function fornon-orthogonal clusters can be also described by the same thresholdfunction Q in Equation (1), where the periods of the grid, H₁ and H₂,are defined in the two directions, perpendicular to the non-orthogonalgrid lines. Similar to the orthogonal case, choosing different gridlines within a parallel set of lines as the distance reference will notchange the values of the dot shape function.

Additional vectors can be used in defining the halftone screen grid. Avector of interest, V₃ is the one that connects an intersection pointwith a second closest neighboring grid point. The spatial vector V₃ isdefined as the summation of V₁ and V₂, or:V ₃ =V ₁ +V ₂.  (2)For an orthogonal screen, this vector is longer than V₁ and V₂ by thesquare root of 2, and is angled 45° from those vectors. As the angles ofthe grid depart from orthogonal, the length of V₃ becomes closer to V₁and V₂. As shown in the hexagonal halftone screen 800 in FIG. 8, as theangle between V₁ and V₂ approaches 120°, for V₁≈V₂ the length of V₃approaches the length of V₁ and V₂ the halftone screen grid 804approaches a regular hexagonal lattice. Each intersection point 802 inthe hexagonal halftone screen grid 804 has 6 nearest neighboringintersection points that are substantially equidistant, and separatedfrom each other in a hexant arrangement.

A third set of parallel lines L₃ crossing intersecting points 802 andparallel to the direction of V₃ can be added to the grid 804, as shownin FIG. 8. The period of the third set of parallel lines is defined asH₃. By vector analysis the three periods, H₁, H₂ and H₃, defined herecan be also described as functions of the three spatial vectors, V₁, V₂and V₃:H ₁ =|V ₂ ×V ₁ |/|V ₁ |=|V ₃ ×V ₁ |/|V ₁|;  (3a)H ₂ =|V ₁ ×V ₂ |/|V ₂ |=|V ₃ ×V ₂ |/|V ₂|;  (3b)H ₃ =|V ₁ ×V ₃ |/|V ₃ |=|V ₂ ×V ₃ |/|V ₃|;  (3c)where V₁×V₂ represents a vector product, or cross product of two vectorsV₁ and V₂, and |V| is the magnitude of the vector V. By the definitionof vector V₃, or Equation (2), it can be shown that the magnitudes ofall vector products shown in Equations (3a)-(3c) are equal and the valueof these magnitudes represents the area A of the parallelograms definedby any two of the three vectors V₁, V₂ and V₃, or:A=H ₁ |V ₁ |=H ₂ |V ₂ |=H ₃ |V ₃|.  (4)

For the following derivation, we select a intersection point o as anreference point and define the three vectors V₁, V₂ and V₃ as all ofthem share the same origin. Then, the distances from an arbitrary pointp to the three spatial vectors V₁, V₂ and V₃ can be described by thefollowing equations, where the spatial vector v is defined as from thereference point o to the arbitrary point p:h ₁ =|v×V ₁ |/|V ₁|  (5a)h ₂ =|v×V ₂ |/|V ₂|  (5b)h ₃ =|v×V ₃ |/|V ₃|  (5c)

Using Equation (4), we can rewrite Equations (5a)-(5c) as:ĥ ₁ ≡h ₁ /H ₁ =|v×V ₁ |/A  (6a)ĥ ₂ ≡h ₂ /H ₂ =|v×V ₂ |/A  (6b)ĥ ₃ ≡h ₃ /H ₃ =|v×V ₃ |/A  (6c)where ĥ₁, ĥ₂, and ĥ₃ are heights normalized by the grid periods H₁, H₂and H₃, respectively and have non-negative values.

We have found a new dot shape function for hexagonal shapedclustered-dot screens:Q=cos [2π(h ₁ /H ₁)]+cos [2π(h ₂ /H ₂)]+cos [2π(h ₃ /H ₃)]  (7a)orQ=cos(2πĥ ₁)+cos(2πĥ ₂)+cos(2πĥ ₃)  (7b)

Since the vector V₃ is defined as the sum of two vectors V₁ and V₂, thedistance h₃ in Equation (7a) and the corresponding normalized height ĥ₃in Equation (7b) are not independent. The vector product in Equation(6c) can be expressed as:v×V ₃ =v×V ₁ +v×V ₂  (8)

In Equation (8) the three vector products are in the same direction butmay carry different signs, plus or minus. Depending on the way ofdefining the spatial vectors, the normalized height ĥ₃ given by Equation(6c) may be equal to either the sum or the difference of ĥ₁ and ĥ₂. Toavoid the ambiguity, we define the two spatial vectors V₁ and V₂, suchthat both vectors start from the reference point o and the angle from V₁to V₂ is positive and the angle is substantially equal to 120°. Thevector V₃ is defined by Equation (2) and all distances used for thenon-orthogonal hexagonal clustered-dot screen are defined by vectorproduct and Equations (5a)-(5c). With these above specified definitions,the following relation between the three normalized heights has beenfound:ĥ ₃ =|ĥ ₂ −ĥ ₁|  (9)

The hexagonal dot shape function in Equation (7b) can be described as afunction of two independent variables ĥ₁ and ĥ₂ as follows:Q=cos(2πĥ ₁)+cos(2πĥ ₂)+cos(2πĥ ₁)cos(2πĥ ₂)+sin(2πĥ ₁)sin(2πĥ ₂)  (10)

Using Equation 7(b), a parametrically controlled hexagonal halftone dotshape function Q₁ can be found as:Q ₁ =a ₁ cos(π(2h ₁ /H ₁))+a ₂ cos(π(2h ₂ /H ₂))+a ₃ cos(π(2h ₃ /H₃))  (11)where the parameter a_(i) is a weight parameter that controls the rateat which the ith vertex of the dot approaches the vertex of aneighboring dot. When the a_(i) are substantially equal, the growth ofthe dot is substantial equal in all 3 vector directions (positive andnegative directions for each of the three vectors). When an a_(i) valueis larger than the other a_(i) values, the dot becomes eccentric,growing at a faster rate in the direction (positive and negativedirections) of the vector associated with the larger a_(i). In thisrespect, this parameter can be used to control which vertices touch atparticular gray levels. For example, the a parameter in Equation (11),(a₁, a₂, a₃) allows neighbor touching at one, two or three differentrespective gray levels, thereby avoiding tone instability caused bytouching all neighbors simultaneously (i.e. at the same gray level).

The dot shape function Q₁ is a parametrically controlled hexagonalhalftone dot shape function which is used to generate the hexagonalhalftone screens for halftoning an image by transforming the image data518 representing image pixels into halftoned image data 524, alsoreferred to as halftone screen output, representing halftoned image dotsthat are printed to form a halftoned image. The parametricallycontrolled hexagonal dot shape function Q₁ (or Q₂ or Q₃ described below)is used to form, or “grow”, a dot to be printed within each halftonecell by darkening pixels in correspondence to the image gray level beingrepresented at that portion of the halftoned image. Halftoned imageobjects, or image regions, having darker gray levels are formed byprinting larger dots, while those having lighter gray levels are formedby printing smaller dots.

Referring to FIG. 9A, an example halftone screen output generated by theparametrically controlled dot shape function Q₁ of Equation (11) isshown producing a halftoned image 900 having a continuously varying graylevel progressing from a minimum (e.g. 0 representing the lowest orlightest gray level) at the left, to a maximum (e.g. 255, or otherscaled number, representing the highest or darkest gray level) at theright. The clustered-dots produced by the parametrically controlled dotshape function Q₁ of Equation (11) begin as substantially circular dots902 at low gray levels, growing to some selected form of hexagon beingsubstantially hexagonal, producing for example regular hexagonal dots904 in FIG. 9A, at mid-gray levels, and closes producing a plurality ofcompact triangular holes 906 between adjacent dots 908 in the darker orhigher gray levels.

It can be desirable to control the contour (shape) of a dot perimeterand the touch points to compensate for attributes of the marking engineused to print the halftoned image. The parametrically controlled dotshape function Q₂ shown in Equation (12) utilizes an additional set ofparameters to control the roundness and convexity/concavity of the dotsides and the sharpness of the vertex touch points.Q ₂ =a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H ₂)^(γ) ² )+a ₃cos(π(2h ₃ /H ₃)^(γ) ³ )  (12)wherein γ_(i) controls the shape of the sides of the dot. In the dotshape function of Equation (12), the perpendicular distance from thepoint of interest p to the closest grid lines 804 is used to determineeach respective h.

Using a dot shape function Q₂ in which γ<1 results in printed hexagonaldots having pincushion (i.e. concave) shaped sides, producing sharpertouch points with adjacent dots, which can improve stability of graytone for marking processes with significant growth. The halftone outputof FIG. 9A illustrates the use of a parametrically controlled dot shapefunction Q₂ with a_(i)=1 and γ_(i)=1.0, FIG. 9B illustrates the use of aparametrically controlled dot shape function Q₂ with a_(i)=1 andγ_(i)=0.6, FIG. 9C illustrates the use of a parametrically controlleddot shape function Q₂ with a_(i)=1 and γ_(i)=0.8.

Using a dot shape function Q₂ having a γ>1 results in a rounder, convexshape, to the printed dot, which can delay (i.e. move) the touching ofadjacent dots to darker gray levels. FIG. 9D illustrates the use of aparametrically controlled dot shape function Q₂ with a_(i)=1 andγ_(i)=1.2, and FIG. 9E illustrates the use of a parametricallycontrolled dot shape function Q₂ with a_(i)=1 and γ_(i)=1.5. In offsetprinting, rounder dots with delayed touching are sometimes used forhalftoning subject matter that is primary at image highlight levels, sothat a relatively round, compact spot with no touching is used for mostof the image.

FIGS. 10A-10I further illustrate how the dot shape function Q₂ can beused to control neighbor halftone dot touching as a function ofhalftoned image gray level. In FIGS. 10A-10I, γ_(i)=1. In FIG. 10A,a₁=a₂=a₃=1.0. In FIG. 10B, a₁=1.25, a₂=1.25, and a₃=1.0. In FIG. 10C,a₁=1.2, a₂=1.0, and a₃=1.0. In FIG. 10D, a₁=1.0, a₂ 1.2, and a₃=1.0. InFIG. 10E, a₁=1.0, a₂=1.0, and a₃=1.25. In FIG. 10F a₁=1.0, a₂=0.8, anda₃=1.0. In FIG. 10G, a₁=0.8, a₂=1.0, and a₃=1.0. In FIG. 10H, a₁=0.8,a₂=1.2, and a₃=1.0. In FIG. 10I, a₁=1.2, a₂=0.8, and a₃=1.0.

Any halftone dot shape function, such as those represented by the dotfunction of Equations (11) and (12) can be inverted, as shown by Q₃ ofEquation (13), to provide an opposite growth sequence, which could bedesirable in some marking processes. More specifically:Q ₃=(a ₁ +a ₂ +a ₃)−(a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H₂)^(γ) ² )+a ₃ cos(π(2h ₃ /H ₃)^(γ) ³ ))  (13)having equation parameters as defined above with respect to Equations(11) and (12). The dot shape function Q₃ provides an interesting compactgrowth sequence from dark triangles at lower gray levels, to darkhexagons as midtones, to light round holes between adjacent dots in thedarker or higher gray levels. As with the un-inverted spot function Q1,Q2, this inverted function Q₃ can be scaled and offset to a desiredrange (e.g., 0 to 255).

The hexagonal dot shape function can be rotated within a fixed positionof the hexagonal grid. More specifically, it is contemplated rotating anentire screen (e.g. the frequency vectors used in generating the screen)along with a dot shape function. But additionally, the present dot shapefunction(s) Q₁, Q₂, and Q₃ can be rotated independently of the screengeometry. That is, dot shape function rotation can be performed by firstdefining the hexagonal screen geometry, which can be expressed invarious equivalent forms using spatial vectors, cells, or frequencyvectors, as described in US Publication Nos. 2008/0130055 and2008/0130054 to Wang, et al., and U.S. Pat. No. 6,798,539 to Wang, etal., mentioned above. Next, a coordinate rotation of the hexagonal dotshape function relative to the screens is performed. That can beaccomplished by defining the screen geometry using coordinates (x, y)and using (x′, y′) in Equation (11), (12) or (13), where:x′=x cos(θ)−y sin(θ)  (14)y′=x sin(θ)+y cos(θ)  (15)and θ is the rotation angle between the dot and the screen orientation.Applying this extension to the dot shape function in Equations (11),(12) or (13) could require some editing of the resulting threshold arrayto avoid spurious dots for some rotation angles, is so desired.

FIGS. 11A-11G illustrate how the dot shape function Q₂ can be rotatedrelative to the screen geometry in this manner, as shown by gray wedgeshaving varying gray levels which were printed using the dot shapefunction rotated by different amounts. In FIG. 11A θ=0°. In FIG. 11Bθ=5°. In FIG. 11C θ=10°. In FIG. 11D θ=15°. In FIG. 11E θ=20°. In FIG.11F θ=30°. In FIG. 11G θ=45°.

The present apparatus and method utilize a parametrically controlledhexagonal halftone dot shape function Q₁, Q₂ and Q₃ that providesoptimum dot touch points as well as compact growth. The touch pointswhich are generated can prevent a tone reproduction bump, while thecompact growth through the gray ranges ensures maximum stability. Themethod defines a threshold function Q₁, Q₂ and Q₃ using the weighted sumof 3 cosines that are functions of distance from 3 respective referencelines defined by the sides of the hexagon. Algebraic powers of thedistances control the shape of the sides of the dot and the sharpness ofeach spot touch point with its neighboring dots. Weights of the cosinescontrol a sequencing of touches, such that contact with the neighboringdots can occur at three different gray levels thereby avoiding a largeinstability that can occur for simultaneous touching. Dot rotation canalso be performed by rotating the coordinate system.

It will be appreciated that various of the above-disclosed and otherfeatures and functions, or alternatives thereof, may be desirablycombined into many other different systems or applications. Also thatvarious presently unforeseen or unanticipated alternatives,modifications, variations or improvements therein may be subsequentlymade by those skilled in the art which are also intended to beencompassed by the following claims.

The invention claimed is:
 1. A method of halftoning which transformsimage data representing contone image pixels into halftoned image datarepresenting halftone dots of a halftoned image, the method comprising:receiving contone image data including pixel values representinggray-scale color densities and pixel location information; andhalftoning the image using an image processor generating clustered-dothalftone screen output representing halftone dots by comparing the imagepixel values to a threshold function in the form of a parametricallycontrolled hexagonal dot shape function, wherein the hexagonal dot shapefunction includes weight parameters controlling the rate at which arespective vertex of a halftone dot approaches a vertex of a neighboringhalftone dot in relation to gray level.
 2. The method of halftoning ofclaim 1 wherein the parametrically controlled hexagonal dot shapefunction isQ ₁ =a ₁ cos(π(2h ₁ /H ₁))+a ₂ cos(π(2h ₂ /H ₂))+a ₃ cos(π(2h ₃ /H ₃))where H₁, H₂ and H₃ are the periods of respective sets of parallel linesintersecting to define a hexagonal halftone screen lattice, h₁, h₂ andh₃ are perpendicular distances from a point of interest in the image toa respective one of the parallel lines from each set, and a₁ a₂ and a₃are weight parameters controlling the rate at which a respective vertexof a halftone dot approaches a vertex of a neighboring halftone dot inrelation to gray level, the halftone dots being substantially circularin halftoned image highlights having lower gray levels, substantiallyhexagonal in halftoned image midtones, and having triangular holesbetween adjacent dots in halftoned image shadows having higher graylevels.
 3. The method of claim 1 wherein the parametrically controlledhexagonal dot shape function isQ ₂ =a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H ₂)^(γ) ² )+a ₃cos(π(2h ₃ /H ₃)^(γ) ³ ) where γ_(i) controls the shape of the sides ofthe halftone dots, H₁, H₂ and H₃ are the periods of respective sets ofparallel lines intersecting to define a hexagonal halftone screenlattice, h₁, h₂ and h₃ are perpendicular distances from a point ofinterest in the image to a closest respective one of the parallel linesfrom each set, and a_(l) a₂ and a₃ are weight parameters controlling therate at which a respective vertex of a halftone dot approaches a vertexof a neighboring halftone dot in relation to gray level, the halftonedots being substantially circular in halftoned image highlights havinglower gray levels, substantially hexagonal in halftoned image midtones,and having triangular holes between adjacent dots in halftoned imageshadows having higher gray levels.
 4. The method of claim 3 whereinγ_(i) is greater than 1 producing halftone screen output resulting inprinted hexagonal dots having concave-shaped sides, thereby producingsharper touch points with adjacent halftone dots, or less than 1producing halftone screen output resulting in printed hexagonal dotshaving convex-shaped sides.
 5. The method of claim 3 wherein Q₂ isscaled to have the same range as the image data gray levels representedby the pixel values.
 6. The method of claim 3 further comprisingperforming dot shape function rotation independently of halftone screengeometry by defining the halftone screen geometry using coordinates (x,y) and using (x′, y′) in Q₂ where:x′=x cos(θ)−y sin(θ)y′=x sin(θ)+y cos(θ) and θ is the rotation angle between the dot and thescreen orientation.
 7. The method of claim 3 further comprisingrendering a halftoned image with a print engine by printing halftonedots having pixels darkened in accordance with their threshold values torepresent the appropriate gray scale level of the contone image.
 8. Themethod of claim 1 wherein the parametrically controlled hexagonal dotshape function is:Q ₃=(a ₁ +a ₂ +a ₃)−(a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H₂)^(γ) ² )+a ₃ cos(π(2h ₃ /H ₃)^(γ) ³ )) where γ_(i) controls the shapeof the sides of the halftone dots, H₁, H₂ and H₃ are the periods ofrespective sets of parallel lines intersecting to define a hexagonalhalftone screen lattice, h₁, h₂ and h₃ are perpendicular distances froma point of interest in the image to a closest respective one of theparallel lines from each set, and a₁ a₂ and a₃ are weight parameterscontrolling the rate at which a respective vertex of a halftone dotapproaches a vertex of a neighboring halftone dot in relation to graylevel, providing a compact growth sequence from dark triangles at lowergray levels, to dark hexagons as midtones, to light round holes betweenadjacent dots in the darker or higher gray levels.
 9. The method ofclaim 8 further comprising performing dot shape function rotationindependently of halftone screen geometry by defining the halftonescreen geometry using coordinates (x, y) and using (x′, y′) in Q₃ where:x′=x cos(θ)−y sin(θ)y′=x sin(θ)+y cos(θ) and θ is the rotation angle between the dot and thescreen orientation.
 10. An image halftoner which transforms image datarepresenting contone image pixels into halftoned image data representinghalftone dots of a halftoned image comprising: an image processorreceiving contone image data including pixel values representinggray-scale color densities and pixel location information, the imageprocessor generating clustered-dot halftone screen output of halftonedots by comparing the image pixel values to a threshold function in theform of a parametrically controlled hexagonal dot shape function,wherein the hexagonal dot shape function includes weight parameterscontrolling the rate at which a respective vertex of a halftone dotapproaches a vertex of a neighboring halftone dot in relation to graylevel.
 11. The image halftoner of claim 10 wherein the parametricallycontrolled hexagonal dot shape function isQ ₁ =a ₁ cos(π(2h ₁ /H ₁))+a ₂ cos(π(2h ₂ /H ₂))+a ₃ cos(π(2h ₃ /H ₃))where H₁, H₂ and H₃ are the periods of respective sets of parallel linesintersecting to define a hexagonal halftone screen lattice, h₁, h₂ andh₃ are perpendicular distances from a point of interest in the image toa respective one of the parallel lines from each set, and a₁ a₂ and a₃are weight parameters controlling the rate at which a respective vertexof a halftone dot approaches a vertex of a neighboring halftone dot inrelation to gray level, the halftone dots being substantially circularin halftoned image highlights having lower gray levels, substantiallyhexagonal in halftoned image midtones, and having triangular holesbetween adjacent dots in halftoned image shadows having higher graylevels.
 12. The image halftoner of claim 10 wherein the parametricallycontrolled hexagonal dot shape function isQ ₂ =a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H ₂)^(γ) ² )+a ₃cos(π(2h ₃ /H ₃)^(γ) ³ ) where γ_(i) controls the shape of the sides ofthe halftone dots, H₁, H₂ and H₃ are the periods of respective sets ofparallel lines intersecting to define a hexagonal halftone screenlattice, h₁, h₂ and h₃ are perpendicular distances from a point ofinterest in the image to a closest respective one of the parallel linesfrom each set, and a₁ a₂ and a₃ are weight parameters controlling therate at which a respective vertex of a halftone dot approaches a vertexof a neighboring halftone dot in relation to gray level, the halftonedots being substantially circular in halftoned image highlights havinglower gray levels, substantially hexagonal in halftoned image midtones,and having triangular holes between adjacent dots in halftoned imageshadows having higher gray levels.
 13. The image halftoner of claim 12wherein γ_(i) is greater than 1 producing halftone screen outputresulting in printed hexagonal dots having concave-shaped sides, therebyproducing sharper touch points with adjacent halftone dots, or γ_(i) isless than 1 producing halftone screen output resulting in printedhexagonal dots having convex-shaped sides.
 14. The image halftoner ofclaim 12 wherein Q₂ is scaled to have the same range as the image datagray levels represented by the pixel values.
 15. The image halftoner ofclaim 12 performing dot shape function rotation independently ofhalftone screen geometry by defining the halftone screen geometry usingcoordinates (x, y) and using (x′, y′) in Q₂ where:x′=x cos(θ)−y sin(θ)y′=x sin(θ)+y cos(θ) and θ is the rotation angle between the dot and thescreen orientation.
 16. The image halftoner of claim 10 wherein theparametrically controlled hexagonal dot shape function is:Q ₃=(a ₁ +a ₂ +a ₃)−(a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H₂)^(γ) ² )+a ₃ cos(π(2h ₃ /H ₃)^(γ) ³ )) providing a compact growthsequence from dark triangles at lower gray levels, to dark hexagons asmidtones, to light round holes between adjacent dots in the darker orhigher gray levels.
 17. The image halftoner of claim 16 performing dotshape function rotation independently of halftone screen geometry bydefining the halftone screen geometry using coordinates (x, y) and using(x′, y′) in Q₃ where:x′=x cos(θ)−y sin(θ)y′=x sin(θ)+y cos(θ) and θ is the rotation angle between the dot and thescreen orientation.
 18. A printing device which transforms image datarepresenting contone image pixels into halftoned image data representinghalftone dots of a halftoned image comprising: an image processorreceiving contone image data including pixel values representinggray-scale color densities and pixel location information, the imageprocessor generating clustered-dot halftone screen output of halftonedots by comparing the image pixel values to a threshold function in theform of a parametrically controlled hexagonal dot shape function,wherein the hexagonal dot shape function includes weight parameterscontrolling the rate at which a respective vertex of a halftone dotapproaches a vertex of a neighboring halftone dot in relation to graylevel; and a print engine rendering a halftoned image by printinghalftone dots having pixels darkened in accordance with their thresholdvalues to represent the appropriate gray scale level of the contoneimage.
 19. The printing device of claim 18 wherein the parametricallycontrolled hexagonal dot shape function isQ ₁ =a ₁ cos(π(2h ₁ /H ₁))+a ₂ cos(π(2h ₂ /H ₂))+a ₃ cos(π(2h ₃ /H ₃))where H₁, H₂ and H₃ are the periods of respective sets of parallel linesintersecting to define a hexagonal halftone screen lattice, h₁, h₂ andh₃ are perpendicular distances from a point of interest in the image toa respective one of the parallel lines from each set, and a₁ a₂ and a₃are weight parameters controlling the rate at which a respective vertexof a halftone dot approaches a vertex of a neighboring halftone dot inrelation to gray level, the halftone dots being substantially circularin halftoned image highlights having lower gray levels, substantiallyhexagonal in halftoned image midtones, and having triangular holesbetween adjacent dots in halftoned image shadows having higher graylevels.
 20. The printing device of claim 18 wherein the parametricallycontrolled hexagonal dot shape function isQ ₂ =a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H ₂)^(γ) ² )+a ₃cos(π(2h ₃ /H ₃)^(γ) ³ ) where γ_(i) controls the shape of the sides ofthe halftone dots, H₁, H₂ and H₃ are the periods of respective sets ofparallel lines intersecting to define a hexagonal halftone screenlattice, h₁, h₂ and h₃ are perpendicular distances from a point ofinterest in the image to a closest respective one of the parallel linesfrom each set, and a₁ a₂ and a₃ are weight parameters controlling therate at which a respective vertex of a halftone dot approaches a vertexof a neighboring halftone dot in relation to gray level, the halftonedots being substantially circular in halftoned image highlights havinglower gray levels, substantially hexagonal in halftoned image midtones,and having triangular holes between adjacent dots in halftoned imageshadows having higher gray levels.
 21. The printing device of claim 20wherein γ_(i) is greater than 1 producing halftone screen outputresulting in printed hexagonal dots having concave-shaped sides, therebyproducing sharper touch points with adjacent halftone dots, or n is lessthan 1 producing halftone screen output resulting in printed hexagonaldots having convex-shaped sides.
 22. The printing device of claim 20performing dot shape function rotation independently of halftone screengeometry by defining the halftone screen geometry using coordinates (x,y) and using (x′, y′) in Q₂ where:x′=x cos(θ)−y sin(θ)y′=x sin(θ)+y cos(θ) and θ is the rotation angle between the dot and thescreen orientation.
 23. The printing device of claim 18 wherein theparametrically controlled hexagonal dot shape function is:Q ₃=(a ₁ +a ₂ +a ₃)−(a ₁ cos(π(2h ₁ /H ₁)^(γ) ¹ )+a ₂ cos(π(2h ₂ /H₂)^(γ) ² )+a ₃ cos(π(2h ₃ /H ₃)^(γ) ³ )) where γ_(i) controls the shapeof the sides of the halftone dots, H₁, H₂ and H₃ are the periods ofrespective sets of parallel lines intersecting to define a hexagonalhalftone screen lattice, h₁, h₂ and h₃ are perpendicular distances froma point of interest in the image to a closest respective one of theparallel lines from each set, and a₁ a₂ and a₃ are weight parameterscontrolling the rate at which a respective vertex of a halftone dotapproaches a vertex of a neighboring halftone dot in relation to graylevel, providing a compact growth sequence from dark triangles at lowergray levels, to dark hexagons as midtones, to light round holes betweenadjacent dots in the darker or higher gray levels.